Theorem opnnei 23149

Description: A set is open iff it is a neighborhood of all of its points. (Contributed by Jeff Hankins, 15-Sep-2009.)

Assertion

Ref	Expression
opnnei	⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))

Distinct variable groups:   𝑥,𝐽   𝑥,𝑆

Proof of Theorem opnnei

Step	Hyp	Ref	Expression
1		0opn 22931	. . . . 5 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
2	1	adantr 480	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → ∅ ∈ 𝐽)
3		eleq1 2832	. . . . 5 ⊢ (𝑆 = ∅ → (𝑆 ∈ 𝐽 ↔ ∅ ∈ 𝐽))
4	3	adantl 481	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → (𝑆 ∈ 𝐽 ↔ ∅ ∈ 𝐽))
5	2, 4	mpbird 257	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → 𝑆 ∈ 𝐽)
6		rzal 4532	. . . 4 ⊢ (𝑆 = ∅ → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
7	6	adantl 481	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
8	5, 7	2thd 265	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
9		opnneip 23148	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
10	9	3expia 1121	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑥 ∈ 𝑆 → 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
11	10	ralrimiv 3151	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
12	11	ex 412	. . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝐽 → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
13	12	adantr 480	. . 3 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (𝑆 ∈ 𝐽 → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
14		df-ne 2947	. . . . . 6 ⊢ (𝑆 ≠ ∅ ↔ ¬ 𝑆 = ∅)
15		r19.2z 4518	. . . . . . 7 ⊢ ((𝑆 ≠ ∅ ∧ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
16	15	ex 412	. . . . . 6 ⊢ (𝑆 ≠ ∅ → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
17	14, 16	sylbir 235	. . . . 5 ⊢ (¬ 𝑆 = ∅ → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
18		eqid 2740	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
19	18	neii1 23135	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑆 ⊆ ∪ 𝐽)
20	19	ex 412	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ⊆ ∪ 𝐽))
21	20	rexlimdvw 3166	. . . . 5 ⊢ (𝐽 ∈ Top → (∃𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ⊆ ∪ 𝐽))
22	17, 21	sylan9r 508	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ⊆ ∪ 𝐽))
23	18	ntrss2 23086	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
24	23	adantr 480	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
25		vex 3492	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
26	25	snss 4810	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ((int‘𝐽)‘𝑆) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆))
27	26	ralbii 3099	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆))
28		dfss3 3997	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ ((int‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ 𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆))
29	28	biimpri 228	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
30	29	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ ∀𝑥 ∈ 𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
31	27, 30	sylan2br 594	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
32	24, 31	eqssd 4026	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) = 𝑆)
33	32	ex 412	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆) → ((int‘𝐽)‘𝑆) = 𝑆))
34	25	snss 4810	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑆 ↔ {𝑥} ⊆ 𝑆)
35		sstr2 4015	. . . . . . . . . . . . . 14 ⊢ ({𝑥} ⊆ 𝑆 → (𝑆 ⊆ ∪ 𝐽 → {𝑥} ⊆ ∪ 𝐽))
36	35	com12 32	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ ∪ 𝐽 → ({𝑥} ⊆ 𝑆 → {𝑥} ⊆ ∪ 𝐽))
37	36	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ({𝑥} ⊆ 𝑆 → {𝑥} ⊆ ∪ 𝐽))
38	34, 37	biimtrid 242	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑥 ∈ 𝑆 → {𝑥} ⊆ ∪ 𝐽))
39	38	imp 406	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ 𝑥 ∈ 𝑆) → {𝑥} ⊆ ∪ 𝐽)
40	18	neiint 23133	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ {𝑥} ⊆ ∪ 𝐽 ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
41	40	3com23 1126	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ {𝑥} ⊆ ∪ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
42	41	3expa 1118	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ {𝑥} ⊆ ∪ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
43	39, 42	syldan 590	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ 𝑥 ∈ 𝑆) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
44	43	ralbidva 3182	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
45	18	isopn3 23095	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆))
46	33, 44, 45	3imtr4d 294	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ∈ 𝐽))
47	46	ex 412	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ ∪ 𝐽 → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ∈ 𝐽)))
48	47	com23 86	. . . . 5 ⊢ (𝐽 ∈ Top → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑆 ⊆ ∪ 𝐽 → 𝑆 ∈ 𝐽)))
49	48	adantr 480	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑆 ⊆ ∪ 𝐽 → 𝑆 ∈ 𝐽)))
50	22, 49	mpdd 43	. . 3 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ∈ 𝐽))
51	13, 50	impbid 212	. 2 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
52	8, 51	pm2.61dan 812	1 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976  ∅c0 4352  {csn 4648  ∪ cuni 4931  ‘cfv 6573  Topctop 22920  intcnt 23046  neicnei 23126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-top 22921  df-ntr 23049  df-nei 23127

This theorem is referenced by:  neiptopreu  23162  flimcf  24011